Vakhtang Paatashvili

Haseman's Problem in Classes of Functions Representable by the Cauchy Type Integral with Density from $L^{p(\cdot)}(\Gamma;\rho)$

abstract:
Haseman's problem $\phi^{+}(\al(t))=a(t)\phi^{-}(t)+b(t)$ is investigated in the class of the Cauchy type integrals with density from the weighted Lebesgue space with a variable exponent $p(t)$. Besides that the problem is studied in a more general statement than that considered earlier, for the first time we cover the case where the boundary condition is prescribed on a non-smooth curve, namely, on a curve with the chord condition.

Mathematics Subject Classification: 47B38, 30E20, 30E25, 42B20, 45P05

Key words and phrases: Curves with a chord-arc-condition, Lebesgue space with a variable exponent, Haseman's boundary value problem, Cauchy type integral with density in $L^{p(\cdot)}(\Gm;\rho)$